A First Course in FUZZY and NEURAL CONTROL

4.3. STABILITY OF FUZZY CONTROL SYSTEMS 145

For a given pair(x(k),u(k)),thenextstatex(k+1)is obtained from the
aboverulebaseasfollows:LetAji(xi(k)) =wji(k)be the membership degree
ofxi(k)in the fuzzy setAji. Using the product t-norm for fuzzy intersection
(ìandî), the membership degree ofx(k)in ruleRjis

vj(k)=

Qn

i=1

wji(k)

Then

x(k+1)=

Pr

j=1vj(kP)[Ajx(k)+Bju(k)]

r

j=1vj(k)

(4.6)

The Takagi-Sugeno fuzzy dynamical model is specified by the matricesAj,j=
1 , 2 ,...,rand the fuzzy setsAji,i=1, 2 ,...,n;j=1, 2 ,...,r.
The open-loop system (no control inputu)is

x(k+1)=

Pr

j=1Pvj(k)Ajx(k)

r

j=1vj(k)

(4.7)

which is nonlinear.
Note that it suffices to study the stability of (4.7) since once a control law
u(∑)is designed, the closed-loop system will be of the same form. Recall that the
equilibrium of the system (4.7) is asymptotically stable in the large ifx(k)→ 0
ask→+∞, for any initialx(0).

Theorem 4.1 (Tanaka and Sugeno)Asufficient condition for the equilib-
rium of the system (4.7) to be asymptotically stable in the large is the existence of
a common positive definite matrixPsuch thatATjPAj< 0 for allj=1, 2 ,...,r

(whereATj denotes the transpose ofAj).

Proof. Since (4.7) is nonlinear, the theorem will be proved if we canfind a
Lyapunov function for the system, that is, a scalar functionV(x(k))such that

1.V(0) = 0

2.V(x(k))> 0 forx(k) 6 =0

3.V(x(k))→∞askx(k)k→∞

4. 4 V(x(k)) =V(x(k+1))−V(x(k))< 0
   Consider the functionV defined by
   V(x(k)) =x(k)
   T
   Px(k)

Then (1), (2), and (3) are easy to verify. For (4), we have

4 V(x(k)) =

1

Pr

j=1

Pr

i=1vj(k)vi(k)





Xr

j=1

v^2 j(k)x(k)T

£

ATjPAj−P

§

x(k)

+

X

1 ≤i<j≤r

vi(k)vj(k)x(k)T

£

ATiPAj+ATjPAi− 2 P

§

x(k)



